# Different derivations of the value of \$\prod_{0\leq j

Let $$\eta=e^{\frac{2\pi i}n}$$, an $$n$$-th root of unity. For pedagogical reasons and inspiration, I ask to see different proofs (be it elementary, sophisticated, theoretical, etc) for the following product evaluation.

If $$T(n)=\frac{(3n-2)(n-1)}2$$ and $$i=\sqrt{-1}$$ then $$\prod_{j